We now apply the tests derived in the previous sections to the returns on the ten US sectoral stock indices from Datastream.12 Specifically, our data consists on daily excess returns for the period January 4th, 1988 - October 12th, 2007 (4971 observations), where we have used the Eurodollar overnight interest rate as safe rate (Datastream code ECUSDST). The model used is a generalisation of the one in the previous section
12Namely, Basic Materials, Consumer Goods, Consumer Services, Financials, Health Care, Industrials, Oil and Gas, Technology, Telecommunications and Utilities.
(see (27)), in which the mean dynamics are captured by a diagonal VAR(1) model with drift, and the covariance dynamics by a conditionally heteroskedastic single factor model in which the conditional variances of both common and specific factors follow GQARCH(1,1) processes to allow for leverage effects (see Sentana, 1995).
We have estimated this model under three different conditional distribution assump-tions on the standardised innovaassump-tionsε∗t: Gaussian, Student t and GH. We first estimated the model by Gaussian PML and then computed the sup LM and Kuhn-Tucker normality tests described in section 3.4, whose asymptotic and parametric bootstrap p-values are reported in Table 1b. In this sense, it worth mentioning that the skewness component of Mardia’s test would have 220 degrees of freedom in our ten dimensional application.
Our tests show that skewness and excess kurtosis are both very significant, although the kurtosis component is one order of magnitude larger than the skewness test. Neverthe-less, the skewness test is affected by the presence of excess kurtosis. We can control for this effect by using the scaling factor in (26) in estimating the appropriate weighting matrix in (11). When we carry out this adjustment, the skewness test turns out to be 18.10, with a p-value of 0.053. Therefore, while the evidence about the presence excess kurtosis is clearly overwhelming, skewness is less important once we take into account excess kurtosis.
In order to shed further light on this issue, we estimated a multivariate Student t model using the analytical formulae for the score that FSC provide. The results in Table 1a show that the estimate for the tail thickness parameter η, which corresponds to slightly more than 10 degrees of freedom, is significantly larger than 0. Then, on the basis of the Student t ML estimates, we have computed the statistics τkT (¯πT) and τaT(¯πT) introduced in section 4. The results in Table 1c show that we can reject the Student t assumption at conventional levels because of the value we obtain for the skew-ness component τaT (¯πT). However, the one-sided version of the ψ component of the test is unable to reject the Student t specification against the alternative hypothesis of symmetric GH innovations because 1(¯sψψT(¯πT, 1, 0) > 0) = 0.
Finally, we re-estimated the model under the assumption that the conditional dis-tribution of the innovations is GH using the analytical expressions for the score that we derive in Appendix B.2. In this case, the GH log-likelihood introduces as additional parameters ψ and the ten-dimensional vector b. Since the ML estimate of ψ reported in
Table 1a is 1, and ˆηT is positive, the estimated GH conditional distribution effectively corresponds to an asymmetric t.
The KT results are confirmed by the LR tests. Specifically, the LR test of Gaus-sian versus symmetric Student t innovations yields a value of 2246.63, which is highly significant, despite being more than four times smaller than the corresponding KT test.
Note, though, that the asymptotic equivalence of the KT and LR tests only holds under the null of Gaussianity and sequences of local alternatives, which is clearly not the case in the data. Similarly, the LR test of Student t vs. asymmetric GH innovations also rejects the null, although the gains in fit obtained by allowing for asymmetry are not as important as those previously obtained by generalising the normal distribution in the leptokurtic dimension. This fact is also likely to explain why the LR and KT test are now commensurate. Interestingly, the asymptotic and bootstrapped p-values are fairly similar in all cases.
Conceivably, though, the rejection of the null hypotheses of normal and Student t innovations that we find could be exacerbated by misspecification of the first and especially second conditional moments. If our specification of the model dynamics is correct, however, the marked distributional differences that we have found should not affect the consistency of the Gaussian PML estimators of θ. With this in mind, we compare the multivariate Gaussian estimate of the conditional variance with the one obtained with a univariate model for the equally weighted portfolio. Specifically, the univariate model is a Gaussian AR(1)-GQARCH(1,1) model. Reassuringly, Figure 8a shows that the (log)standard deviations of the two series display a very similar pattern, although the univariate estimates are somewhat noisier. An alternative check of our dynamic specification is to compare the multivariate Gaussian and GH estimates of the covariance matrix. The Gaussian estimates should remain consistent even in the absence of normality, while the GH ones will only be consistent if this distribution is correctly specified. Figure 8b shows the (log)standard deviations of the equally weighted portfolio that we obtain with these two distributions, which again display a very similar pattern.
As mentioned in the introduction, one of the main reasons for taking into account departures from normality is to avoid biases in the estimation of the quantiles of the distribution, which are required for instance in V@R calculations. To determine to what extent the Student t and the GH distributions are more useful than the Gaussian
assumption across all conceivable quantiles, we have computed the empirical cumulat-ive distribution function of the one-period-ahead probability integral transforms of the equally weighted portfolio of Datastream indices generated by the three fitted distribu-tions (see Diebold, Gunther, and Tay, 1998). Figure 9a shows the difference between the corresponding cumulative distributions and the 45 degree line. Under correct specifica-tion, those differences should tend to 0 asymptotically. Unfortunately, a size-corrected version of the usual Kolmogorov or Cramer von Mises tests that takes into account the sample uncertainty in the estimation of the underlying parameter estimates is rather difficult to obtain in this case. Nevertheless, the graph clearly indicates that the Student t and asymmetric t distributions clearly provide a better fit than the normal. At the same time, the two non-normal distributions are hard to distinguish.
Since these results are specific to one particular portfolio, we repeat the same ex-ercise for 5,000 different portfolios whose weights are randomly chosen from a uniform distribution on the unit sphere. Figure 10, which compares the empirical distribution function across those 5,000 portfolios of the Cramer von Mises goodness of fit statistics, clearly shows that the symmetric and asymmetric t dominate the normal distribution in the first-order stochastic sense. Once again, though, it is difficult to distinguish the two fat tailed distributions. In order to magnify the differences between the symmetric and asymmetric t distributions, we look at the most asymmetric portfolio, whose weights are proportional to b (see Menc´ıa and Sentana, 2008, for a formal justification). In this sense, Figure 9b shows that the asymmetric t provides a slightly better fit to the em-pirical distribution of this benchmark portfolio. Note, however, that even in this case the gains from allowing for asymmetry are smaller than those from taking into account excess kurtosis.
Finally, an interesting question is whether the results that we have obtained are specific to the daily frequency. We assess this issue by repeating our tests for weekly returns of the same 10 Datastream US indices in excess of the Eurodollar weekly interest rates, with the same parametrisation of the conditional mean vector and covariance matrix. Table 2 shows the results. Again, normality is easily rejected, mainly because of the presence of excess kurtosis.13 In this case, though, the Student t distribution cannot be rejected at the 5% level, although the LR tests are significant at the 10% level. This
13The skewness score based test becomes 12.42 once we adjust the weighting matrix in (11) by assuming ellipticity but not Gaussianity.
can be due to the smaller sample size (1043 observations). Notice also that the LR test of asymmetric t vs. asymmetric GH innovations is positive but not significant at conventional levels.
7 Conclusions
In this paper, we propose LM and LR specification tests of multivariate normality and multivariate Student t against alternatives with GH innovations, which is a rather flexible multivariate asymmetric distribution that also nests as particular cases many other well known and empirically realistic examples. Methodologically, our main contribution is to explain how to overcome the identification problems that the use of the GH distribution as an embedding model entails. We derive closed form expressions for the score based tests and decompose our proposed statistics into skewness and kurtosis components.
From these expressions, we obtain more powerful one-sided KT versions and show their asymptotic equivalence to LR tests. For this reason, we would recommend the KT instead of the LM tests. We also exploit this equivalence to obtain the common asymptotic distributions of the LR and KT tests, which turn out to be standard despite the non-standard features of the problem.
We assess the finite sample size properties of the testing procedures that we propose and previously suggested methods by means of detailed Monte Carlo exercises. Our results indicate that the asymptotic sizes of our normality tests are very reliable in finite samples. However, we also find that the kurtosis component of the Student t test is too conservative, and the same is true of the corresponding LR test. Nevertheless, we show that one can correct those distortions by means of a parametric bootstrap, although obtaining reliable p-values for the LR test is computationally time consuming. In finite samples, we find that the LR and KT tests yield very similar power in both the Gaussian and Student t cases for parameter configurations that cannot be regarded as local to the null.
Finally, we present an empirical application to the ten US sectoral daily excess stock returns from Datastream. We can easily reject normality because the skewness and es-pecially kurtosis components of our tests are highly significant. And while a multivariate symmetric Student t seems to fit well the kurtosis of the data, the skewness component of the Student t is still significant. In sum, our results suggest that the conditional
distribution of the returns on those US indices is mildly asymmetric and strongly lep-tokurtic. From a risk management perspective, the quantiles of arbitrary portfolios are much better captured by either the Student t or the GH distribution than by the normal for most cases, while the GH distribution displays a slightly better fit than the Student t in the case of the most asymmetric portfolio. We also compute tests for the same data at the weekly frequency, finding similar results, although in this case asymmetry seems to be even less important, which may be due to the smaller sample size.
An interesting extension of our results would be to test multivariate normality against a general location-scale mixture of normals, although the resulting tests will also be affected by the same type of underidentification problems under the null. Alternatively, we could consider as our null hypothesis other special cases of the GH distribution, such as the symmetric normal-gamma. Finally, one could use the test statistics that we have derived to improve the efficiency of indirect estimators along the lines suggested by Calzolari, Fiorentini, and Sentana (2004).
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